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If tuvw = 108

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adkikani
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If tuvw = 108 [#permalink] New post  06 Dec 2017, 06:51
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If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A. 1
B. 16
C. 17
D. 18
E. 19
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Re: If tuvw = 108 [#permalink] New post  06 Dec 2017, 08:05
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Factorize 108 = 1 * 2^2 * 3^3

t > u > v > w --> w = 1; v = 2; u = 3; t = 2 * 3^2 = 18

t - w = 18 - 1 = 17

Answer: C
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generis
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If tuvw = 108 [#permalink] New post  06 Dec 2017, 09:36
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adkikani wrote:
If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A. 1
B. 16
C. 17
D. 18
E. 19

List the prime factors of 108: 2*2*3*3*3

To maximize the difference between the greatest number and the smallest number, make \(t\) as big as possible by making \(w\), \(v\) and \(u\) as small as possible -- remembering t > u > v > w

Let \(w\) = 1
Let \(v\) = 2
Let \(u\) = 3

From prime factor list, the remaining numbers are 2, 3, and 3
Let \(t\) = (2 * 3 * 3) = 18
(t - w) = (18 - 1) = 17

Answer C
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Re: If tuvw = 108 [#permalink] New post  20 Oct 2019, 20:27
Clearly tuvw are factors of 108. To find the greatest possible value of t-w we need to maximise t and minimise w

Lets look at the factors of 108
1-108
2-54
3-36
4-27
6-18
9-12

We need 4 factors and W is our minimum
Choose w=1
1-108
2-54
3-36
4-27
6-18
9-12

In the right hand column choose a value for t, you can test numbers but you'll see that 18 is the only one that works as
So 18-1 = 17
1-108
2-54
3-36
4-27
6-18
9-12
We can process eliminate the answers by asking whether we can obtain a given value.
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Re: If tuvw = 108 [#permalink] New post  22 Nov 2020, 13:36
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adkikani wrote:
If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A. 1
B. 16
C. 17
D. 18
E. 19


The prime factorization of \(108 = 2^2 * 3^3\).

We're told \(w < v < u < t\). Using the factors, we need to create 4 different integers.

We can get the numbers \(1 < 2 < 3 < 18\)

\(18 - 1 = 17\).

Answer is C.
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Re: If tuvw = 108 [#permalink] New post  22 Nov 2020, 14:12
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To MAXIMIZE the difference ,we need to minimize w.
w is a factor of 108 and the minimum such a factor is 1.Hence w=1.
108=3*3*3*2*2*1
So w=1 (smallest)
Next smallest v=2
and u=3 in the same way.
So, w, v, u are minimized and t is maximized if w=1,v=2,u=3.
This means t=18
[ t=(108/3*2)=108/6=18 ]
So t-w =18-1=17 (C)

ALTERNATELY ,you can also use options in the following way:
Eliminate (a) because t-w cannot be 1 (As t>u>v>w and belong to integers)
Now add 1 to the options and see which of them is a factor of 108.
Eliminate (b),(d) and (e)
Only (c) which is 17 when added to 1 gives 18 that is a factor of 108.
Hence (c)
Hope this helps! Keep studying!:)
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Abhishek009
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Re: If tuvw = 108 [#permalink] New post  28 Mar 2024, 08:07
 
adkikani wrote:
If tuvw = 108, and t, u, v, and w are all positive integers such that t > u > v > w, what is the greatest possible value for t - w?

A. 1
B. 16
C. 17
D. 18
E. 19

\(­108 = 1*2^2*3^3\)

So, \(t = 18\) , \(u = 3\) , \(v = 2\) & \(w = 1\)

Hence, \(t - w = 18 - 1 => 17\), Answer will be (C)
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Re: If tuvw = 108 [#permalink]
28 Mar 2024, 08:07
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